What is an intuitive explanation of the combinations formula?

Say you draw three cards from a deck: 2 of clubs, 4 of hearts, 6 of diamonds.

Draw them again, but in a different order: 4 of hearts, 6 of diamonds, 2 of clubs.

It's the same hand. You just rearranged the order of the slots.

Or, put another way, once you draw these three cards, you can rearrange them in any order you like (there are $3!$ ways to do so), but it's still the same single combination of three cards.

For each three-card hand you can draw, you can draw the same three cards in six different orderings. Dividing by $6$, therefore, "collapses" each specific ordering of three cards down to a single set of those three cards.

The specific orderings (permutations) total

$$_{52}P_3 = \frac{52!}{(52-3)!}$$

but the combinations total

$$_{52}C_3 = \frac{52!}{(52-3)! \cdot 3!} = \frac{_{52}P_3}{3!}.$$